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University of Johannesburg Faculty of Engineering and the Built Environment Mining Department 2 Module: Geotechnical Engineering 2B Code: GEMINB2 Section: Rock Mechanics Contents: Rockmass Behaviour & Response to Excavation Geological Deformation of Rocks Rock Failure Criteria Prepared By: WB Motlhabane Next Revision Date: January 2019 Table of Contents 1 INTRODUCTION ............................................................................................................ 4 2 TERMINOLOGY ............................................................................................................ 5 2.1 PRIMITIVE STRESS................................................................................................................. 5 2.2 PRINCIPAL STRESS ................................................................................................................ 5 2.3 INDUCED STRESS .................................................................................................................. 5 2.4 FIELD STRESS ....................................................................................................................... 5 2.5 SPAN ................................................................................................................................. 5 2.6 CONVERGENCE .................................................................................................................... 5 2.7 CLOSURE ............................................................................................................................ 5 2.8 CRITICAL HALF-SPAN ............................................................................................................. 6 3 CONVERGENCE ............................................................................................................ 7 4 CRITICAL HALF-SPAN ..................................................................................................... 8 5 AVERAGE CONVERGENCE................................................................................................ 9 6 ENERGY RELEASE RATE ................................................................................................ 10 6.1 DEFINITION OF ERR ........................................................................................................... 10 6.2 CALCULATION OF ERR. ....................................................................................................... 10 6.3 SIGNIFICANCE OF ERR ........................................................................................................ 10 6.4 REDUCING ERR ................................................................................................................. 11 6.4.1 STABILISING PILLARS 11 6.4.2 BACKFILL 11 7 STRESS AROUND STOPING EXCAVATIONS .......................................................................... 12 7.1 INTRODUCTION .................................................................................................................. 12 7.2 INDUCED STRESS ................................................................................................................ 12 7.3 CLOSURE AND CONVERGENCE .............................................................................................. 14 7.4 STRESS DISTRIBUTION AROUND A STOPE ................................................................................. 15 7.4.1 ZONES OF SHEAR STRESS 15 7.4.2 ZONES OF HIGH COMPRESSIVE STRESS 15 7.4.3 TENSILE ZONE 16 7.4.4 ZONE OF LOW STRESS 16 8 FRACTURING AROUND STOPES ....................................................................................... 17 8.1 INTRODUCTION .................................................................................................................. 17 8.2 FRACTURE GEOMETRY AROUND A STOPE ................................................................................ 17 9 STRESS AROUND TUNNELS ............................................................................................ 19 9.1 INTRODUCTION .................................................................................................................. 19 9.2 STRESS AROUND A RECTANGULAR TUNNEL.............................................................................. 19 9.3 STRESS AROUND A CIRCULAR TUNNEL .................................................................................... 20 9.4 CHANGING STRESS AROUND TUNNELS .................................................................................... 21 9.5 STRESS AROUND A CIRCULAR OPENING................................................................................... 21 9.5.1 CALCULATION OF TANGENTIAL STRESS 22 9.5.2 EFFECTS OF TANGENTIAL STRESS 25 10 FRACTURING AROUND TUNNELS ..................................................................................... 26 11 GEOLOGICAL DEFORMATION OF ROCKS ........................................................................ 27 11.1 BRITTLE-DUCTILE PROPERTIES OF THE LITHOSPHERE .............................................................. 27 11.2 DEFORMATION IN PROGRESS ............................................................................................. 27 © University of Johannesburg Page 2 of 48 11.3 EVIDENCE OF PAST DEFORMATION ..................................................................................... 27 11.4 FAULTS........................................................................................................................... 30 11.4.1 TYPES OF FAULTS 30 11.5 EVIDENCE OF MOVEMENT ON FAULTS ................................................................................. 33 11.6 FOLDS AND TOPOGRAPHY.................................................................................................. 38 11.7 FOLDS AND METAMORPHIC FOLIATION................................................................................ 39 11.8 MOUNTAINS AND MOUNTAIN BUILDING PROCESSES ............................................................. 39 12 ROCK FAILURE CRITERIA............................................................................................... 43 12.1 GRIFFITH THEORY ............................................................................................................ 43 12.2 MOHR- COULOMB FAILURE CRITERION ............................................................................... 45 12.3 HOEK AND BROWN FAILURE CRITERION ............................................................................... 45 13 REFERENCES.............................................................................................................. 48 List of figures FIGURE 1 ENERGY RELEASE RATE .................................................................................................................... 11 FIGURE 2 TERMINOLOGY............................................................................................................................... 12 FIGURE 3 STOPE STRESS DISTRIBUTION............................................................................................................. 13 FIGURE 4 STOPE CLOSURE ............................................................................................................................. 14 FIGURE 5 STRESS DISTRIBUTION...................................................................................................................... 15 FIGURE 6 “STRESS LINES”.............................................................................................................................. 16 FIGURE 7 STOPE FRACTURE GEOMETRY ............................................................................................................ 17 FIGURE 8 FACE PARALLEL FRACTURES .............................................................................................................. 18 FIGURE 9 TUNNEL STRESS DISTRIBUTION .......................................................................................................... 19 FIGURE 10 RECTANGULAR TUNNEL ................................................................................................................. 20 FIGURE 11 CIRCULAR TUNNEL STRESS .............................................................................................................. 21 FIGURE 12 CHANGING TUNNEL STRESS ............................................................................................................ 21 FIGURE 13 RADIAL & TANGENTIAL STRESS ........................................................................................................ 22 FIGURE 14 TANGENTIAL STRESS...................................................................................................................... 23 FIGURE 15 STRESS CURVES FOR DIFFERENT K VALUES .......................................................................................... 23 FIGURE 16 STRESS DISTRIBUTION AT CIRCLE BOUNDARIES .................................................................................... 25 FIGURE 17 SPALLED BOREHOLE, INDICATING STRESS ORIENTATION ........................................................................ 25 FIGURE 18 TUNNEL FRACTURING .................................................................................................................... 26 FIGURE 19 "GOTHIC ARCH?" ......................................................................................................................... 26 © University of Johannesburg Page 3 of 48 1 Introduction The crust of the earth is subject to a certain magnitude and direction of stress. This is as a result of the force exerted on the rock mass by the weight of the overlying strata as well as some other “primitive” sources of stress, such as: Tectonic forces, and Major geological structures, etc. When excavations are made during mining operations, these primitive stress patterns are disturbed and modified, sometimes to magnitudes much higher that the original. The stress added to the primitive stresses is called “Induced stress”. Mining excavations therefore induce additional stress on the rock mass surrounding them as well as on geological structures in the area. When the total stress exceeds the inherent strength of the rock mass or the cohesive strength of seismically active faults and dykes, stress induced failure can lead to instability of the rock mass. This instability can manifest in the form of: Unstable block of rock between fracture planes subject to gravity - resulting in rock falls, Failure of a beam subject to tensile stress, Rock bursts from a rock mass failing under compression, and Rock bursts from shear stress induced on a fault plane or shear fracture. A good understanding of the conditions leading to these instabilities is required. © University of Johannesburg Page 4 of 48 2 Terminology 2.1 Primitive stress This is the stress that exists is the rock mass prior to the event of any mining activity. It is there because of natural phenomena, such as: Gravity, Tectonic forces, and Geological structure. 2.2 Principal Stress This is the largest component of stress that exists in a given rock mass. It is a vector, i.e. it has magnitude and direction. Major principal stress is usually denoted by the symbol 1 is expressed in the SI unit Pa (Pascal) or multiples of Pa (kPa, MPa, etc.). 2.3 Induced Stress Induced stress is the magnitude of stress by which the primitive stress is modified by the existence of excavations in the rock mass. The SI units are the same as for principal stress. 2.4 Field Stress Field Stress or Total Stress or Absolute Stress is the sum of all significant Induced Stresses and the Primitive Stress. Stress components of any magnitude or sign can be added together, provided that their orientations correspond. Field stress is usually denoted by the symbol q. 2.5 Span Span is the distance between two diverging stope faces (one advancing “east” and one advancing “west”) in a narrow tabular stope. The half span is taken as half this distance and is denoted by the symbol L. The SI unit for this term is m. 2.6 Convergence Convergence is the reduction in the width of an excavation due to the elastic deformation of the rock mass. In a narrow tabular stope, convergence will be the reduction in distance between the hanging wall and footwall (i.e. reduction in stoping width) due to the elastic deformation of the rock mass. This figure can be calculated from theory and is a function of: Elasticity of the rock mass, Depth below surface, and Span. The SI unit of convergence is m or mm. 2.7 Closure Closure is a term used to describe the combined elastic and inelastic components of deformation of an excavation. This figure has been measured to be up to five times as much as the theoretical convergence and cannot be modelled accurately at depth because of the large number of variables involved. Closure rates are determined from observation and average values are assigned to specific ground control districts. Closure can be expressed as a function of face advance (mm/m) or as function of time (mm/day). © University of Johannesburg Page 5 of 48 2.8 Critical half-span The critical half-span for a stope is half the distance between the two diverging faces when the stope has advanced to a position that has caused a convergence at the centre of equal to the stope width. It is denoted by the symbol Lc and the SI unit is m. © University of Johannesburg Page 6 of 48 3 Convergence To calculate the convergence at a given point in a stope that has two diverging long wall faces, the following equation is used: Sz 2(1 v)q G L2 X 2 ................................................................................................ Equation 1 Where: v G q L X Poisson’s ratio (no unit) Modulus of Rigidity (Pa) Virgin stress before mining (ρgh) (Pa) Half span (m) The distance from the stope center to the point being considered (m) The modulus of rigidity is derived from E and v as follows: G E ................................................................................................................ Equation 2 2(1 v) Where: G E V Modulus of rigidity (Pa) Young’s Modulus (Pa) Poisson’s ratio (no unit) © University of Johannesburg Page 7 of 48 4 Critical half-span The critical half-span for a stope is half the distance between the two diverging faces when the stope has advanced to a position that has caused a convergence at the centre of equal to the stope width. (The hanging wall and footwall have come together). This number can be calculated by substituting the stope width for convergence in the convergence equation and then resolving for Lc. This produces the following equation: Lc S mG 2(1 v)q .............................................................................................................. Equation 3 Where: Lc Sm G v q Critical half span (m) Stope width (m) Modulus of rigidity (Pa) Poisson’s ratio (no unit) Virgin stress before mining (ρgh) (Pa) © University of Johannesburg Page 8 of 48 5 Average convergence It is necessary to determine the average converge of a stope if one wants to calculate the Energy Release Rate. There are three different calculations, depending on the magnitude of L (Half-span). The following equations apply: Save π(1 v)Lq 2G when 0 L L c ............................................................................... Equation 4 Save 0,79Sm when L Lc ...................................................................................... Equation 5 © University of Johannesburg Page 9 of 48 6 Energy Release Rate 6.1 Definition of ERR ERR is a measure of the energy released during convergence and redistribution of stresses per unit area of excavation, measured in MJ/m2. During the process of mining, rock is removed from the rock mass. This results in energy changes due to the sag of the overlying strata and the redistribution of stress from the mined to the unmined areas in the rock mass. This causes stress to be concentrated ahead of mining faces as well as in remnants and lagging corners of the face. The Energy Release Rate is a calculated number that provides a measure of these energy changes and stress concentrations. 6.2 Calculation of ERR. ERR = Stress x Average Convergence, therefore the following equations will apply: ERR π(1 v)Lq 2G 2 when 0 L L c ............................................................................ Equation 6 ERR 0,79Sm q when L Lc .................................................................................... Equation 7 6.3 Significance of ERR Although ERR is calculated on a model based on a perfectly elastic situation, its relevance is not limited to a perfectly elastic rock mass only. In the South African hard rock environment, the rock mass can generally be regarded as elastic with a fracture zone of 5 to 10% of the mined area. For this situation, the way in which ERR is calculated makes it acceptably accurate and relevant. Energy release rate can therefore be used to describe the mining environment in understandable terms and assist in designing appropriate layouts and support systems for specific mining areas. ERR correlates well with the following: Length of span, Stresses ahead of the stope face, Shear Stresses on planes of weakness, Depth and height of fracturing, Fracture dilation and thrust on the hanging wall beam, Closure, and Incidence of rock bursts. When considering ERR in mine design, there is no general rule of thumb. The critical ERR level could vary strongly from one area to another and is dependent on the geotechnical conditions of each area. In general, a figure in excess 40 MJ/m2 would be regarded as indicative of hazardous conditions, however, in specific circumstances, a much lower number might be required. The graph in Figure 1 (After Jager & Ryder) shows the massive difference in ERR levels between two different reefs in the same area. It is evident that the target ERR level for the VCR should be much lower than for the Carbon Leader. © University of Johannesburg Page 10 of 48 Rock bursts / 1000 m2 stoped 4 Ventersdorp Contact Reef Carbon Leader Reef 3 2 1 0 10 20 30 40 50 60 70 80 90 Energy Release Rate (MJ/m2) Figure 1 Energy Release Rate 6.4 Reducing ERR In areas where high Energy Release Rates become problematic, every effort should be made to reduce the ERR to an acceptable level. How is this done? ERR is directly proportional to Stress and Convergence, i.e. ERR increases as Stress and/or Convergence increase and decrease as Stress and/or Convergence decrease. Not much can be done to decrease Stress, but there are definite ways of controlling and decreasing Convergence. The most generally used methods of reducing convergence in South Africa are Backfill and Stabilising Pillars as well as combinations of both. 6.4.1 Stabilising Pillars These are blocks of ground left unmined on the reef horizon in a predetermined pattern that will reduce the span and control the convergence. They can be arranged either on dip or on strike. The ERR is directly proportional to the percentage extraction, i.e. the larger the Percentage extraction (less pillars) the larger the ERR. 6.4.2 Backfill Regional support can be accomplished by placing backfill into the back areas of a stope. Depending on the quality and timing of backfill, it can have a significant impact on the Energy Release Rate. © University of Johannesburg Page 11 of 48 7 Stress around stoping excavations 7.1 Introduction Before a stoping excavation is made, the prevailing stress would be equal to primitive stress and would mainly be a function of: Depth, Rock density, Geological structure, and Tectonic forces. The resultant lateral stress is a function of the vertical stress and the magnitude of k. For the sake of simplicity, primitive stress will be calculated on the assumption that gravity is the only primitive source and therefore, the following equations will apply: qv ρgh ............................................................................................................ Equation 8 qh kq v ...................................................................................................................... Equation 9 Where: qv qh ρ g h Vertical virgin stress (Pa) Horizontal virgin stress (Pa) Overburden density (kg/m3) Gravitational acceleration (m/s2) – 9,81m/s2 for Rock Mechanics Depth (m) 7.2 Induced stress As soon as an excavation is made in a rock mass, the primitive stress field will be altered. The modifying effect of the excavation on the original stress (in both magnitude and direction) is called the induced stress. This effect is usually and amplification (in other words, the stress will increase significantly). The direction of stress may also change. To illustrate the principles involved, the example of a horizontal section through a narrow tabular stope will be used; see Figure 2 and note the terminology: Half span (L) Stope Width Sm West face East face Stope center Figure 2 Terminology Before this stope excavation was made, the stress field (primitive) in the rock mass would have been largely subject to gravtity and therefore 1 would have been vertical and a function of depth and rock density. The stoping excavation has now created a void. The stress will therefore be diverted to the rock mass on either side of the excavation, the east and west © University of Johannesburg Page 12 of 48 faces, setting up compressive stresses on these abutments that will be higher than the original. See Figure 3. West face East face Figure 3 Stope stress distribution The dotted lines represent the primitive stress field before the excavation was made and the solid lines represent the stress field modified by the excavation. Note how the stress is displaced to the abutments, setting up zones of high compressive stress ahead of each stope face (east and west). The magnitude of the stress on and ahead of each stope face will be a function of the depth, the span (or half span), rock density and the distance ahead of the face. Theoretically, the zone of largest stress will be right on the face and the magnitude will then diminish as a function of distance ahead of the face, until, some distance away; it will be reduced to a value very close to primitive stress. The magnitude stress ahead of the face is calculated from theory, using the formulae below: σy qX X2 L2 σ x q(k 1) ....................................................................................................................... Equation 10 qX X2 L2 ...................................................................................................... Equation 11 Where: σy σx L X q Vertical stress (Pa) Horizontal stress (Pa) Half span (m) Distance from stope center to point considered (m) Virgin vertical stress before mining (Pa) The stress at the face can obviously not be calculated with these (X = L and dividing by zero will not present a solution). Therefore, the “root mean square” method should be used to calculate the stress on the face, namely: © University of Johannesburg Page 13 of 48 σRMS 2,51q L ...................................................................................................... Equation 12 Sm Where: σRMS q L Sm Vertical stress on the face (Pa) Virgin vertical stress before mining (Pa) Half span (m) Stope width (m) The table below shows the stress distribution on and ahead of the face at a depth of 2 000m and a stope width of 1,1m: Distance ahead of face (m) y MPa 0 1 2 5 10 25 1266 377 269 174 127 88 The UCS of quartzite usually ranges from around 180Mpa to 220Mpa – note that the stress levels in the immediate face vicinity is far in excess of this number and therefore it is certain that the rock mass will fail in some way in this region. The rock mass will, if fact, fail at stresses well below the UCS (usually at magnitudes of 30 to 60% of UCS). This is the reason why a zone of fractured rock will extend to about 10m ahead of the stope face. As soon as the ground fractures, the stress in that area is relieved and restored to a number very close to q. The zone of highest stress is then shifted into the unfractured zone just ahead. As the face advances, the fracture zone will propagate and the stress peak will move forward by the same distance. The magnitude of stress is a function of the depth and span, therefore, the bigger the span and depth, the larger the stress and the more intense the fracturing around the stope will be. 7.3 Closure and convergence As the span is increased by face advance, the rock mass will deform, both elastically and inelastically. The hanging wall and footwall will converge (move closer together) as the excavation attempts to “close up”. The elastic component of the deformation is called “convergence” and the sum of the elastic and inelastic deformation is called “closure”. Maximum closure will take place in the stope center (half – way between the east and west faces). This implies that the hanging wall beam (and to a certain extent the footwall beam) will be subject to a bending deformation and therefore under tension. See Figure 4. Tension West face East face Closure Figure 4 Stope closure © University of Johannesburg Page 14 of 48 7.4 Stress distribution around a stope Four distinctly different zones of stress may develop around a narrow tabular stoping excavation (See Figure 5). Zone of high compressive stress ahead of the face, Tensile zone above the stope, Zone of shear stress above and below the face, and Zone of low stress below the stope. - + + Figure 5 Stress distribution Shear stress zones 7.4.1 + High compressive stress zones - Tensile zone Low stress zone Zones of Shear Stress The shear stress that develops above and below (as well as ahead of) the face is a result of the differential in stress as the high compressive stress close to the face diminishes with distance. If these shear stresses sufficiently exceed the cohesive strength of a discontinuity that they intersect, a violent shear failure may occur. Events such as these have caused large magnitude rock bursts in deep level gold mines in South Africa. 7.4.2 Zones of High Compressive Stress Extremely high compressive stresses on the abutments of stopes may lead to intense fracturing and also to smaller magnitude rock bursts. The resultant lateral strain caused by these high stresses result in: Dilation of the stope face (bulging into the excavation), and Lateral forces in the hanging wall and footwall. At medium stress conditions, these lateral forces have the effect of “clamping” the blocks of rock together in the hanging wall, thereby affording some “natural” stability. They are then referred to as “clamping forces”. When they become excessive, they may cause the hanging wall beam to fail due to buckling. © University of Johannesburg Page 15 of 48 Intersection of the stress zones of two adjacent excavations lead to a compounding effect whereby the sum of the two zones lead to extremely high stresses, such as a pillar between two converging stope faces or a remnant. 7.4.3 Tensile Zone The hanging wall beam is subject to tension due to closure as well as compression by lateral stress. In low stress shallower conditions, the tension will by far exceed the compression and the hanging wall beam may be at risk of tensile failure. If a large discontinuity is mined through, the tensile stress across it can cause separation and result in large-scale rock falls. In these cases, the stabilizing effect of clamping forces is also neutralised. In situations where the lateral compression exceeds the tension, stability is ensured by the effect of clamping forces and the risk of tensile beam failure is small. The extent of the tensile zone above a stope usually diminishes with depth. 7.4.4 Zone of Low Stress The zone below a stope is effectively de-stressed as the original stress is diverted to the abutments. The magnitude of stress will be a function of the depth below the stope. The stress lines can be seen as “wrapping” vertically around the stope face and curling back underneath the stope at an angle of around 45. See Figure 6. “Stress lines” Abutment Excavations that are placed behind these “stress 45 lines” will be subject to low stresses whereas those placed on the outside may become very highly Figure 6 “Stress lines” stressed. This principle is referred to as the “45 shadow”. Mines with high stress regimes therefore endeavour to keep footwall tunnels within this zone. This principle is used on deep level gold mines where the footwall drives are kept behind the face as well as in diamond mines doing block caving, where the production level is kept behind the undercut level. This is usually referred to as “Follow-on development”. The zone of low stress could also present a hazard when excavations that were previously subjected to high stress levels are de-stressed as the stope face advances across them. The fractured rock mass around such excavations is suddenly “relaxed” and this could cause instability of blocks of rock that were stabilized through the clamping effect of the previously higher stress. © University of Johannesburg Page 16 of 48 8 Fracturing around stopes 8.1 Introduction As seen in the previous chapter, a stoping excavation may induce compression, tension and shear stress in the rock mass surrounding it. The magnitude of these stresses is a function of: Depth, Span, and Proximity to other excavations. When the level of stress exceeds the inherent strength of the rock mass, failure is to be expected and this will manifest in the formation of compression, shear and tensile fracturing of the rock mass. Observations made in many of the deeper level gold mines of stress induced fracturing have shown that this failure usually occurs in a very specific geometry. The depth and intensity of fracturing may vary as a result of the depth and the rock mass qualities. The description of the fracture geometry described in this chapter is a very generalized one and fracturing may vary from one area to another. 8.2 Fracture geometry around a stope The fracture zone around a stope may be as deep as 20m above and below the stope and extend to around 10m ahead of the stope face. Figure 7 illustrates the typical geometry and types of fractures that may be formed. Shear Fractures Primary Extension Fractures Flat Fractures Bedding planes Stope Secondary Extension Fractures Flat dipping Fractures Figure 7 Stope fracture geometry The table that follows describes all the features shown in Figure 7. © University of Johannesburg Page 17 of 48 Primary extension fractures Secondary extension fractures Shear Fractures Flat Fractures Flat dipping fractures Occur at the extremity of the fracture zone and are probably caused by deformation of the fracture zone as the face advances. Are formed within 2m of the face when intact pieces of rock between primary fractures fail. Near vertical ahead of the face, but curl back in the hanging wall to angles of about 70°. These develop at the extremity of the fracture zone where total stress > UCS. Hanging wall and footwall fractures are independent mirror images. Last to appear. Generally of limited extent and parallel to hanging wall. Patches of shallow dipping fractures (20° to 40°) could form in “hard patches” near to the stope face. They are sporadic and only a few metres long. Stope face From the section it was seen that the fractures are generally steeply dipping, but that two distinctly different sets of fractures may form that dip in opposite directions. These two set may intersect each other and fragment the hanging wall beam into blocks. As long as they are steeply dipping, stability is usually insured due to the presence of clamping forces. The fractures generally mimic the excavation shape and wraps around any changes in the direction of the face, such as sidings and headings. Where they change direction, the dip of the fractures tends to become flatSteeply dipping ter and stability could then be probfractures (70) lematic. See Figure 8. When a straight face shape is maintained and the strike distance between panels (lead or lag) is kept to a minimum, the fractures tend to be generally steep and parallel to the face. These are usually fairly stable and easy to support. Irregular face shape and excessive leads or lags may cause intersection between steep and shallow fractures and fragment the rock mass into blocks. Blocky ground carries a higher risk of instability and requires more intensive support systems to ensure stability. © University of Johannesburg Excavation Solid Flat dipping fractures (20 - 30) Figure 8 Face parallel fractures Page 18 of 48 9 Stress around tunnels 9.1 Introduction When a tunnel is excavated, the same principle applies as described in the previous chapter. The field stress in the rock mass will be altered and diverted to the sides of the tunnel. Depending on the shape of the tunnel and the orientation of the field stress, its walls may become subjected to high compressive stresses in some areas under tension in other areas. Tunnels placed in close proximity to stoping excavations as well as other excavations will seldom be subjected to a vertically aligned field stress and therefore it should not be assumed that the sidewalls would exclusively be the areas of high stress concentration. Figure 9 illustrates the distribution of stress around a tunnel for various directions of field stress. q Tension High compression High compression Tension High compression q q = Vertical Tension Tension High compression q slanted @ 45 Figure 9 Tunnel stress distribution The shape of a tunnel has a major influence on the stress profile induced in the surrounding rock mass as well as the stability of a tunnel. A circular shape will generally induce a lower stress profile than a rectangular shape and for that reason, mines that have high stress conditions generally favour and arched shape tunnel to keep the compressive stresses on the tunnel walls as low as possible. Very frequently, a tunnel will, over its lifespan be subjected to a changing stress regime because of changing position of the stope face over a period of time. 9.2 Stress around a rectangular tunnel Stresses around tunnels with a rectangular cross section are very complex to define mathematically due to the extreme sensitivity that these stresses have for the degree of rounding of © University of Johannesburg Page 19 of 48 the corners. Theoretically, there will be infinite stress at a true rectangular corner, but in practice, the rock in and around the corners becomes crushed and as a result, the effective shape of the excavation is rectangular with more or less rounded corners. Rectangular tunnels induce much higher levels of stress on the surrounding rock mass and the near rectangular corners are then subjected to a high degree of fracturing with resultant high risk of instability. Figure 10 illustrates the perimeter stress around a rectangular profile. Note that is the direction of q. Excavation Profile Figure 10 Rectangular tunnel 9.3 Stress around a circular tunnel The stress induced by a circular tunnel is lower, being a maximum of 2q (for k=1) and 3q (for k=0) on the perimeter of the tunnel as shown in Figure 11. © University of Johannesburg Page 20 of 48 q 3q 2q q 2R 3R 4R 5R Figure 11 Circular tunnel stress Note that the magnitude of the stress is a maximum on the tunnel wall and then decays back to a value of q about 2.5 times the width away from the tunnel. 9.4 Changing stress around tunnels High stress around a tunnel is not the only source of stress-related instability; the changes in stress (both direction and magnitude) can incur significant damage to a tunnel. Changes is stress usually occur as the stope face is advanced across the tunnel. See Figure 12. As a stope face is advanced across a tunnel, it could be subjected to a vertical q at A, slanted and higher stress at B and C and the finally de-stressed at D. Stope face D C Direction of face advance B A Figure 12 Changing tunnel stress 9.5 Stress around a circular opening Stresses around circular openings are generally shown either as radial stresses or as tangential stresses. Radial stress (σr) has a direction that can be extended through the origin of the circle. Tangential stress (σθ) has a direction of 90˚ to the radius. See Figure 13. © University of Johannesburg Page 21 of 48 Figure 13 Radial & tangential stress These stresses are calculated with the following equations: σθ σr q 2 q 2 (1 k)(1 (1 R2 q R4 ) (1 k)(1 3 )cos2θ ....................................................... Equation 13 r2 2 r4 R2 q R2 R4 ) (1 k)(1 4 3 )cos2θ ....................................................... Equation 14 r2 2 r2 r4 Where: R r θ q k Radius of the opening (m) Distance from the center of the circle to the point considered (m) Polar angle measured counter clockwise from the horizontal (°) Field stress (Pa) Ratio of horizontal to vertical stress (no unit) In the case of a shaft or tunnel, one would mostly consider the stresses on the circumference, at which point R = r. This simplifies the formulae considerably. Radial stresses on the circumference are always zero, because the equal and opposite stress on the inside is obviously zero. Even away from the circumference, the radial stress is seldom of any great significance. Tangential stresses have a major influence on the stability of tunnels and for the purpose of this course, only these will be considered. 9.5.1 Calculation of tangential stress σθ q 2 (1 k)(1 R2 q R4 ) (1 k)(1 3 )cos2θ r2 2 r4 From this equation it can be seen that the tangential stress is dependent on: , the polar angle © University of Johannesburg Page 22 of 48 q, the original stress k, the ratio between horizontal and vertical stress y x Remember is measured counter clockwise from the X-axis. Figure 14 Tangential stress The cosine of the polar angle can have a maximum value of 1 and a minimum value of -1. The diagram in Figure 14 shows where the maximum values are: From this it can be seen that the maximum values will be at 0˚ and at 180˚, where cos2 has its largest value of 1. The largest tangential stresses are thus at the two points at right angles to the principal stress direction. As mentioned before, when considering stresses on the circumference, R = r, which means in R2 R 4 the above equation 2 4 1 and this simplifies the formula to: r r σθ = 3q-qk ...................................................................................................................... Equation 14 .................................................................................................................................................. The value of k varies between 0 for a uniaxial stress and 1 for a biaxial stress where the two components are equal. Substitution of these values into the simplified equation will result in: For k = 0, σθ = 3q and for k = 1 σθ = 2q. The maximum tangential stress thus ranges between twice and three times the original stress (q) depending on the value of k. Note that the field stress returns to a value of q at a point 5 x R away from the origin of the circle. © University of Johannesburg Figure 15 Stress curves for different k values Page 23 of 48 Figure 15 shows the stress curves for values of k = 0 and k = 1 respectively. Note that the highest stress is on the circumference and then diminishes to a value of q at a distance of 5 radii away from the centre. Note that the smallest values of tangential stress will occur where the value of 2 = -1 and 3q 2q q 2R 3R 4R 5R these are the points where = 90 and = 270. These stresses can be negative (tensile) and could thus have a major effect and should be considered. © University of Johannesburg Page 24 of 48 9.5.2 Effects of tangential stress Where the original stress is vertical or near vertical, the largest tangential stress will occur on the sides of a circular tunnel and the lowest stress will occur in the hanging wall and the footwall. For a uniaxial principal stress, the maximum value is three times that of the original stress and as seen in the previous section, the original stress can already be a very high number induced by stoping in the area. When extremely high stresses occur on the sides of the tunnel, it can result in severe fracturing that could be impossible to support. Where the original stress is near horizontal, extreme conditions might be experienced in the hanging wall and footwall. The points of lowest tangential stress are sometimes subject to tensile stresses. These would endeavour to pull the rock apart, allowing key block to be released. The stress distribution around a circular opening can be graphically represented as shown in Figure 18, which illustrates the distribution for an opening under vertical uniaxial stress. q q -q 3q 3q -q q q Figure 16 Stress distribution at circle boundaries The stress distribution as shown in Figure 16 generally puts the sidewalls of a tunnel under compressive stress, causing fracturing and resultant slabbing of the sidewalls. The hanging wall and footwall might be subject to tensile stresses, causing shear cracks. These could open up and key blocks might dislodge and fall out. These effects can also be witnessed when examining diamond drill holes at depth. The fracturing caused by high tangential stresses manifest as fractured zones called “dog-ears”, see Figure 17. The orientation of these “dog-ears” can indicate the direction principal stress as shown in the same figure. Tunnel failure would typically occur when the induced stresses are extreme, subjecting the sidewalls to slabbing and buckling into the excavation as illustrated in Figure 18. 100MPa 50MPa Figure 17 Spalled borehole, indicating stress orientation © University of Johannesburg Page 25 of 48 10 Fracturing around tunnels High compressive stresses do the most damage to tunnel walls, causing them to be fractured and “slabbed” and then “bulge” or fall into the excavation. The orientation of q will determine where the biggest stress concentration will occur; if q is vertical, the sidewalls will sustain the most damage. If q is horizontal, the hanging wall and footwall may suffer deformation. See Figures 18 & 19. Tensile zone, causing separation and shear cracks Slabbing and buckling of sidewalls due to high compressive stresses Tunnel damage, typical of a vertical q. Original Excavation Figure 18 Tunnel fracturing Fractured zone Note the direction of q and the resultant deformation due to high compression. q Direction of biggest stress Figure 19 "Gothic arch?" © University of Johannesburg Page 26 of 48 11 Geological Deformation of Rocks 11.1 Brittle-Ductile Properties of the Lithosphere We all know that rocks near the surface of the Earth behave in a brittle manner. Crustal rocks are composed of minerals like quartz and feldspar which have high strength, particularly at low pressure and temperature. As we go deeper in the Earth the strength of these rocks initially increases. At a depth of about 15 km we reach a point called the brittle-ductile transition zone. Below this point rock strength decreases because fractures become closed and the temperature is higher, making the rocks behave in a ductile manner. At the base of the crust the rock type changes to peridotite which is rich in olivine. Olivine is stronger than the minerals that make up most crustal rocks, so the upper part of the mantle is again strong. But, just as in the crust, increasing temperature eventually predominates and at a depth of about 40 km the brittle-ductile transition zone in the mantle occurs. Below this point rocks behave in an increasingly ductile manner. 11.2 Deformation in Progress Only in a few cases does deformation of rocks occur at a rate that is observable on human time scales. Abrupt deformation along faults, usually associated with earthquakes occurs on a time scale of minutes or seconds. Gradual deformation along faults or in areas of uplift or subsidence can be measured over periods of months to years with sensitive measuring instruments. 11.3 Evidence of Past Deformation Evidence of deformation that has occurred in the past is very evident in crustal rocks. For example, sedimentary strata and lava flows generally follow the law of original horizontality. Thus, when we see such strata inclined instead of horizontal, evidence of an episode of deformation. Since many geologic features are planar in nature, we a way to uniquely define the orientation of a planar feature we first need to define two terms - strike and dip. © University of Johannesburg Page 27 of 48 For an inclined plane the strike is the compass direction of any horizontal line on the plane. The dip is the angle between a horizontal plane and the inclined plane, measured perpendicular to the direction of strike. In recording strike and dip measurements on a geologic map, a symbol is used that has a long line oriented parallel to the compass direction of the strike. A short tick mark is placed in the center of the line on the side to which the inclined plane dips, and the angle of dip is recorded next to the strike and dip symbol as shown above. For beds with a 900 dip (vertical) the short line crosses the strike line, and for beds with no dip (horizontal) a circle with a cross inside is used as shown below. For linear structures, a similar method is used, the strike or bearing is the compass direction and angle the line makes with a horizontal surface is called the plunge angle. © University of Johannesburg Page 28 of 48 Fracture of Brittle Rocks As we have discussed previously, brittle rocks tend to fracture when placed under a high enough stress. Such fracturing, while it does produce irregular cracks in the rock, sometimes produces planar features that provide evidence of the stresses acting at the time of formation of the cracks. Two major types of more or less planar fractures can occur: joints and faults. Joints As we learned in our discussion of physical weathering, joints are fractures in rock that show no slippage or offset along the fracture. Joints are usually planar features, so their orientation can be described as a strike and dip. They form from as a result of extensional stress acting on brittle rock. Such stresses can be induced by cooling of rock (volume decreases as temperature decreases) or by relief of pressure as rock is eroded above thus removing weight. Joints provide pathways for water and thus pathways for chemical weathering attack on rocks. If new minerals are precipitated from water flowing in the joints, this will form a vein. Many veins observed in rock are mostly either quartz or calcite, but can contain rare minerals like gold and silver. These aspects will be discussed in more detail when we talk about valuable minerals from the earth in a couple of weeks. Because joints provide access of water to rock, rates of weathering and/or erosion are usually higher along joints and this can lead to differential erosion. From an engineering point of view, joints are important structures to understand. Since they are zones of weakness, their presence is critical when building anything from dams to highways. For © University of Johannesburg Page 29 of 48 dams, the water could leak out through the joints leading to dam failure. For highways the joints may separate and cause rock falls and landslides. 11.4 Faults Faults occur when brittle rocks fracture and there is an offset along the fracture. When the offset is small, the displacement can be easily measured, but sometimes the displacement is so large that it is difficult to measure. 11.4.1 Types of Faults As we found out in our discussion of earthquakes, faults can be divided into several different types depending on the direction of relative displacement. Since faults are planar features, the concept of strike and dip also applies, and thus the strike and dip of a fault plane can be measured. One division of faults is between dip-slip faults, where the displacement is measured along the dip direction of the fault, and strike-slip faults where the displacement is horizontal, parallel to the strike of the fault. Recall the following types of faults: Dip Slip Faults - Dip slip faults are faults that have an inclined fault plane and along which the relative displacement or offset has occurred along the dip direction. Note that in looking at the displacement on any fault we don't know which side actually moved or if both sides moved, all we can determine is the relative sense of motion. o Normal Faults - are faults that result from horizontal tensional stresses in brittle rocks and where the hanging-wall block has moved down relative to the footwall block. Horsts & Grabens - Due to the tensional stress responsible for normal faults, they often occur in a series, with adjacent faults dipping in opposite directions. In such a case the down-dropped blocks form grabens and the uplifted blocks form horsts. In areas where tensional stress has recently affected the crust, the grabens may form rift valleys and the uplifted horst blocks may form linear mountain ranges. The East African Rift Valley is an example of an area where continental extension has created such a rift. The basin and range province of the western U.S. (Nevada, Utah, and Idaho) is also an area © University of Johannesburg Page 30 of 48 that has recently undergone crustal extension. In the basin and range, the basins are elongated grabens that now form valleys, and the ranges are uplifted horst blocks. Half-Grabens - A normal fault that has a curved fault plane with the dip decreasing with depth can cause the down-dropped block to rotate. In such a case a half-graben is produced, called such because it is bounded by only one fault instead of the two that form a normal graben. o Reverse Faults - are faults that result from horizontal compressional stresses in brittle rocks, where the hanging-wall block has moved up relative the footwall block. © University of Johannesburg Page 31 of 48 A Thrust Fault is a special case of a reverse fault where the dip of the fault is less than 45o. Thrust faults can have considerable displacement, measuring hundreds of kilometers, and can result in older strata overlying younger strata. Strike Slip Faults - are faults where the relative motion on the fault has taken place along a horizontal direction. Such faults result from shear stresses acting in the crust. Strike slip faults can be of two varieties, depending on the sense of displacement. To an observer standing on one side of the fault and looking across the fault, if the block on the other side has moved to the left, we say that the fault is a left-lateral strike-slip fault. If the block on the other side has moved to the right, we say that the fault is a right-lateral strike-slip fault. The famous San Andreas Fault in California is an example of a right-lateral strike-slip fault. Displacements on the San Andreas fault are estimated at over 600 km. © University of Johannesburg Page 32 of 48 11.5 Evidence of Movement on Faults Since movement on a fault involves rocks sliding past each other there may be left evidence of movement in the area of the fault plane. Fault Breccias are crumbled up rocks consisting of angular fragments that were formed as a result of grinding and crushing movement along a fault. When the rock is broken into clay or silt size particles as a result of slippage on the fault, it is referred to as fault gouge. Slickensides are scratch marks that are left on the fault plane as one block moves relative to the other. Slickensides can be used to determine the direction and sense of motion on a fault. Mylonite - Along some faults rocks are sheared or drawn out by ductile deformation along the fault. This results in a type of localized metamorphism called dynamic metamorphism (also called cataclastic metamorphism. The resulting rock is a fine grained metamorphic rock show evidence of shear, called a mylonite. Faults that show such ductile shear are referred to as shear zones. Deformation of Ductile Rocks When rocks deform in a ductile manner, instead of fracturing to form faults or joints, they may bend or fold, and the resulting structures are called folds. Folds result from compressional stresses or shear stresses acting over considerable time. Because the strain rate is low and/or the temperature is high, rocks that we normally consider brittle can behave in a ductile manner resulting in such folds. Geometry of Folds - Folds are described by their form and orientation. The sides of a fold are called limbs. The limbs intersect at the tightest part of the fold, called the hinge. A line connecting all points on the hinge is called the fold axis. An imaginary plane that includes the fold axis and divides the fold as symmetrically as possible is called the axial plane of the fold. © University of Johannesburg Page 33 of 48 We recognize several different kinds of folds. Monoclines are the simplest types of folds. Monoclines occur when horizontal strata are bent upward so that the two limbs of the fold are still horizontal. © University of Johannesburg Page 34 of 48 Anticlines are folds where the originally horizontal strata has been folded upward, and the two limbs of the fold dip away from the hinge of the fold. Synclines are folds where the originally horizontal strata have been folded downward, and the two limbs of the fold dip inward toward the hinge of the fold. Synclines and anticlines usually occur together such that the limb of a syncline is also the limb of an anticline. In the diagrams above, the fold axes are horizontal, but if the fold axis is not horizontal the fold is called a plunging fold and the angle that the fold axis makes with a horizontal line is called the plunge of the fold. © University of Johannesburg Page 35 of 48 Note that if a plunging fold intersects a horizontal surface, we will see the pattern of the fold on the surface (see also figures 11.15e in your text. Domes and Basins are formed as a result of vertical crustal motion. Domes look like an overturned bowl and result from crustal upwarping. Basins look like a bowl and result from subsidence (see figure 11.14 in your text). Folds are described by the severity of folding. an open fold has a large angle between limbs, a tight fold has a small angle between limbs. Further classification of folds include: If the two limbs of the fold dip away from the axis with the same angle, the fold is said to be a symmetrical fold. © University of Johannesburg Page 36 of 48 If the limbs dip at different angles, the folds are said to be asymmetrical folds. If the compressional stresses that cause the folding are intense, the fold can close up and have limbs that are parallel to each other. Such a fold is called an isoclinal fold (iso means same, and cline means angle, so isoclinal means the limbs have the same angle). Note the isoclinal fold depicted in the diagram below is also a symmetrical fold. If the folding is so intense that the strata on one limb of the fold becomes nearly upside down, the fold is called an overturned fold. An overturned fold with an axial plane that is nearly horizontal is called a recumbant fold. A fold that has no curvature in its hinge and straight-sided limbs that form a zigzag pattern is called a chevron fold. © University of Johannesburg Page 37 of 48 11.6 Folds and Topography Since different rocks have different resistance to erosion and weathering, erosion of folded areas can lead to a topography that reflects the folding. Resistant strata would form ridges that have the same form as the folds, while less resistant strata will form valleys (see figure11.14 in you text). How Folds Form Folds develop in two ways: Flexural folds form when layers slip as stratified rocks are bent. This results in the layers maintaining their thickness as they bend and slide over one another. These are generally formed due to compressional stresses acting from either side. Flow folds form when rocks are very ductile and flow like a fluid. Different parts of the fold are drawn out by this flow to different extents resulting in layers becoming thinner in some places and thicker in outer places. The flow results in shear stresses that smear out the layers. Folds can also form in relationship to faulting of other parts of the rock body. In this case the more ductile rocks bend to conform to the movement on the fault. Also since even ductile rocks can eventually fracture under high stress, rocks may fold up to a certain point then fracture to form a fault. © University of Johannesburg Page 38 of 48 11.7 Folds and Metamorphic Foliation As we saw in our discussion of metamorphic rocks, foliation is a planar fabric that develops in rocks subject to compressional stress during metamorphism. It may be present as flattened or elongated grains, with the flattening occurring perpendicular to the direction of compressional stress. It also results from the reorientation, recrystallization, or growth of sheet silicate minerals so that their sheets become oriented perpendicular to the compressional stress direction. Thus, we commonly see a foliation that is parallel to the axial plane of the fold. Shearing of rock during metamorphism can also draw out grains in the direction of shear. 11.8 Mountains and Mountain Building Processes One of the most spectacular results of deformation acting within the crust of the Earth is the formation of mountain ranges. Mountains frequently occur in elongate, linear belts. They are constructed by tectonic plate interactions in a process called orogenesis. Mountain building (orogenesis) involves Structural deformation. Faulting. Folding. Igneous Processes. Metamorphism. Glaciation. Erosion. Sedimentation Constructive processes, like deformation, folding, faulting, igneous processes and sedimentation build mountains up; destructive processes like erosion and glaciation, tear them back down again. Mountains are born and have a finite life span. Young mountains are high, steep, and growing upward. Middle-aged mountains are cut by erosion. Old mountains are deeply eroded and often buried. Ancient orogenic belts are found in continental interiors, now far away from plate boundaries, but provide information on ancient tectonic processes. Since orogenic continental crust generally has a low density and thus is too buoyant to subduct, if it escapes erosion it is usually preserved. Uplift and Isostasy © University of Johannesburg Page 39 of 48 The fact that marine limestones occur at the top of Mt. Everest, indicates that deformation can cause considerable vertical movement of the crust. Such vertical movement of the crust is called uplift. Uplift is caused by deformation which also involves thickening of the low density crust and, because the crust "floats" on the higher density mantle, involves another process that controls the height of mountains. The discovery of this process and its consequences involved measurements of gravity. Gravity is measured with a device known as a gravimeter. A gravimeter can measure differences in the pull of gravity to as little as 1 part in 100 million. Measurements of gravity can detect areas where there is a deficiency or excess of mass beneath the surface of the Earth. These deficiencies or excesses of mass are called gravity anomalies. A positive gravity anomaly indicates that an excess of mass exits beneath the area. A negative gravity anomaly indicates that there is less mass beneath an area. Negative anomalies exist beneath mountain ranges, and mirror the topography and crustal thickness as determined by seismic studies. Thus, the low density continents appear to be floating on higher density mantle. The protrusions of the crust into the mantle are referred to as crustal roots. Normal crustal thickness, measured from the surface to the Moho is 35 to 40 km. But under mountain belts crustal thicknesses of 50 to 70 km are common. In general, the higher the mountains, the thicker the crust. What causes this is the principal of isostasy. The principal can be demonstrated by floating various sizes of low density wood blocks in your bathtub or sink. The larger blocks will both float higher and extend to deeper levels in the water and mimic the how the continents float on the mantle (see figure 11.26 in your text). © University of Johannesburg Page 40 of 48 It must be kept in mind, however that it's not just the crust that floats, it's the entire lithosphere. So, the lithospheric mantle beneath continents also extends to deeper levels and is thicker under mountain ranges than normal. Because the lithosphere is floating in the asthenosphere which is more ductile than the brittle lithosphere, the soft asthenosphere can flow to compensate for any change in thickness of the crust caused by erosion or deformation. The Principle of isostasy states that there is a flotational balance between low density rocks and high density rocks. i.e. low density crustal rocks float on higher density mantle rocks. The height at which the low density rocks float is dependent on the thickness of the low density rocks. Continents stand high because they are composed of low density rocks (granitic composition). Ocean basins stand low, because they are composed of higher density basaltic and gabbroic rocks. Isostasy is best illustrated by effects of glaciation. During an ice age crustal rocks that are covered with ice are depressed by the weight of the overlying ice. When the ice melts, the areas previously covered with ice undergo uplift. Mountains only grow so long as there are forces causing the uplift. As mountains rise, they are eroded. Initially the erosion will cause the mountains to rise higher as a result of isostatic compensation. But, eventually, the weight of the mountain starts to depress the lower crust and sub-continental lithosphere to levels where they start to heat up and become more ductile. This hotter lithosphere will then begin to flow outward away from the excess weight and the above will start to collapse. The hotter rocks could eventually partially melt, resulting in igneous intrusions as the magmas move to higher levels, or the entire hotter lower crust could begin to rise as a result of their lower density. These processes combined with erosion on the surface result in exhumation, which causes rocks from the deep crust to eventually become exposed at the surface. Causes of Mountain Building There are three primary causes of mountain building. 1. Convergence at convergent plate boundaries. 2. Continental Collisions. 3. Rifting Convergent Plate Margins When oceanic lithosphere subducts beneath continental lithosphere magmas generated above the subduction zone rise, intrude, and erupt to form volcanic mountains. The compressional stresses generated between the trench and the volcanic arc create fold-thrust mountain belts, and similar compression behind the arc create a fold-thrust belt resulting in mountains. Mountains along the margins of western North and South America, like the Andes and © University of Johannesburg Page 41 of 48 the Cascade range formed in this fashion. Island arcs off the coast of continents can get pushed against the continent. Because of their low density, they don't subduct, but instead get accreted to the edge of the continent. Mountain ranges along the west coast of North America formed in this fashion (see figure 11.20 in your text). Continental Collisions Plate tectonics can cause continental crustal blocks to collide. When this occurs the rocks between the two continental blocks become folded and faulted under compressional stresses and are pushed upward to form fold-thrust mountains. The Himalayan Mountains (currently the highest on Earth) are mountains of this type and were formed as a result of the Indian Plate colliding with the Eurasian plate. Similarly the Appalachian Mountains of North America and the Alps of Europe were formed by such processes. Rifting Continental Rifting occurs where continental crust is undergoing extensional deformation. This results in thinning of the lithosphere and upwelling of the asthenosphere which results in uplift. The brittle lithosphere responds by producing normal faults where blocks of continental lithosphere are uplifted to form grabens or half grabens. The uplifted blocks are referred to a fault-block mountains. © University of Johannesburg Page 42 of 48 12 Rock Failure Criteria 12.1 Griffith Theory The Griffiths criterion describes the relationship between applied nominal stress and crack length at fracture, i.e. when it becomes energetically favourable for a crack to grow. Griffith was concerned with the energetics of fracture, and considered the energy changes associated with incremental crack extension. For a loaded brittle body undergoing incremental crack extension, the only contributors to energy changes are the energy of the new fracture surfaces (two surfaces per crack tip) and the change in potential energy in the body. The surface energy term (S) represents energy absorbed in crack growth, while the some stored strain energy (U) is released as the crack extends (due to unloading of regions adjacent to the new fracture surfaces). Surface energy has a constant value per unit area (or unit length for a unit thickness of body) and is therefore a linear function of (crack length), while the stored strain energy released in crack growth is a function of (crack length)2, and is hence parabolic. These changes are indicated in the figure below: The next step in the development of Griffith's argument was consideration of the rates of energy change with crack extension, because the critical condition corresponds to the maximum point in the total energy curve, i.e. dW/da = 0, where a = a*. For crack lengths greater than this value (under a given applied stress), the body is going to a lower energy state, which is favourable, and hence fast fracture occurs. dW/da = 0 occurs © University of Johannesburg Page 43 of 48 when dS/da= dU/da. The sketch below shows these energy rates, or differentials with respect to a. R is the resistance to crack growth (= dS/da) and G is the strain energy release rate (= dU/da). When fracture occurs, R = G and we can define Gcrit as the critical value of strain energy release, and equate this to R. Hence Gcrit represents the fracture toughness of the material. In plane stress the Griffith equation is: where, to get the fracture stress in MPa (the standard SI engineering unit), the critical strain energy release rate is in N/m, E is in N/m2, and a is in m. This provides an answer in N/m2 (Pa), which needs to be divided by 106 to get the standard engineering unit of MPa. In plane strain: © University of Johannesburg Page 44 of 48 12.2 Mohr- Coulomb Failure Criterion The Mohr-Coulomb criterion is the most common failure criterion encountered in geotechnical engineering. Many geotechnical analysis methods and programs require use of this strength model. The Mohr-Coulomb criterion describes a linear relationship between normal and shear stresses (or maximum and minimum principal stresses) at failure. The Mohr-Coulomb criterion implementation in RocData can be used to analyze both direct shear and triaxial test data. The direct shear formulation of the criterion is given by the following Eqn: The Mohr-Coulomb criterion for triaxial data is given by the following Eqn: where c is the cohesive strength, and is the friction angle. 12.3 Hoek and Brown Failure Criterion © University of Johannesburg Page 45 of 48 The Generalized Hoek-Brown criterion is an empirical failure criterion which establishes the strength of rock in terms of major and minor principal stresses. It predicts strength envelopes that agree well with values determined from laboratory triaxial tests of intact rock, and from observed failures in jointed rock masses. RocData implements the most recent update (the 2002 edition) of the Generalized HoekBrown criterion. This edition resolves some formerly troublesome issues including: The applicability of the criterion to very weak rock masses, and The calculation of equivalent Mohr-Coulomb parameters from the Hoek-Brown failure envelope The Generalized Hoek-Brown criterion is non-linear and relates the major and minor effective principal stresses (sigma1 and sigma3) according to the following equation: where: and are the axial (major) and confining (minor) effective principal stresses respectively is the uniaxial compressive strength (UCS) of the intact rock material mb is a reduced value (for the rock mass) of the material constant mi (for the intact rock) s and a are constants which depend upon the characteristics of the rock mass In most cases it is practically impossible to carry out triaxial tests on rock masses at a scale which is necessary to obtain direct values of the parameters in the Generalized Hoek-Brown equation. Therefore some practical means of estimating the material constants mb, s and a is required. According to the latest research, the parameters of the Generalized Hoek-Brown criterion [Hoek, Carranza-Torres & Corkum (2002)], are given by the following equations: where: © University of Johannesburg Page 46 of 48 GSI (the Geological Strength Index) relates the failure criterion to geological observations in the field mi is a material constant for the intact rock the parameter D is a "disturbance factor" which depends upon the degree of disturbance to which the rock mass has been subjected by blast damage and/or stress relaxation. It varies from 0 for undisturbed in situ rock masses to 1 for very disturbed rock masses. © University of Johannesburg Page 47 of 48 13 References JC Jaeger & NG Cook. “Fundamentals of Rock Mechanics”. JA Ryder & AJ Jager. “A Textbook on Rock Mechanics for Tabular Hard Rock Mines”. AJ Jager & JA Ryder. “A Handbook on Rock Engineering Practice for Tabular Hard Rock Mines”. TR Stacey & AW Swart. “Best Practice, Rock Engineering for “other” Mines”. https://www.google.co.za/search?q=mohr+coulomb+failure+criterion&sa=X&rlz=1C1GGRV_enZA751ZA751&tbm=isch&tbo=u&source=univ&ved=0ah UKEwj-y4O719 https://www.rocscience.com/help/rocdata/webhelp/rocdata/Generalized_HoekBrown_Criterion.htm http://www.tulane.edu/~sanelson/eens1110/deform.htm Physical Geology by Steven Earle used under a CC-BY 4.0 international license © University of Johannesburg Page 48 of 48